Strong Double Space-Time Wave Packets from Optical Parametric Amplification

doi:10.21203/rs.3.rs-1768146/v1

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Strong Double Space-Time Wave Packets from Optical Parametric Amplification

https://doi.org/10.21203/rs.3.rs-1768146/v1

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The space-time (ST) wave packet, a new type light source with many new features, has already found applications in weak-field optics, however its generation method limits its energy and therefore its application in strong-field optics. We report that the type-I collinear optical parametric amplification (OPA) is a natural amplifier for ST wave packets, because in it, the ST spectrum (i.e., angle-dependent phase-matching spectrum) of the signal and idler has the same analytical expression as that of ST wave packets under the narrowband approximation. The high gain in a thin-crystal OPA allows for large energy amplification while ensuring unchanged ST correlation. Meanwhile, for the first time, a double ST wave packet containing two collinear-propagating ones of the amplified signal and the generated idler via the nondegenerate OPA is reported, which with short- and long-wavelengths (relative to the degenerate-wavelength) have superluminal and subluminal velocities, respectively. This study realizes energy amplification, wavelength conversion, and velocity switching of ST wave packets and will certainly expand the application in many fields.

The spatiotemporal coupling, as an ever-changing optics tool, recently has been well studied to realize new type light sources for novel optical phenomena and physical applications [1-5]. Two recent representative examples are “flying focus” proposed by A. Sainte-Marie et al. and D.H. Froula et al. [6, 7] and “ST wave packet” proposed by H.E. Kondakci and A.F. Abouraddy [8, 9], which can propagate at any velocity, i.e., higher or lower than the light speed and forward or backward along the optical axis. The former has already been applied for overcoming dephasing in laser wakefield accelerators [10], producing arbitrary velocity ionization waves [11], etc. in strong-field optics, and the latter also has great potentials from optical imaging, microscopy, communication to micro-particle manipulation. Apart from wide-range tunable velocities and accelerations [12-15], ST wave packets have many other features, including diffraction-free [16], tunable dispersion [17, 18], anomalous refraction [19], anomalous ST dispersion effects [20], veiled Talbot effect [21], time diffraction [22], etc., which would be certainly as important as “flying focus” in the strong-field optics. However, producing a ST wave packet needs to accurately control its ST spectrum in the Fourier space to introduce a necessary ST correlation that is transverse spatial-frequency dependent quasi-monochromatic spectral-frequency [23-25], and in this process, a phase spatial light modulator (SLM) is required at the Fourier plane of a grating 4-f setup [9]. The low damage threshold of SLM determines that the ST wave packet generator cannot operate at a high energy level, and the same problem also occurs with other ST optical fields generated by metasurfaces [26-28]. Consequently, an efficient amplifier is required, but which must keep the ST correlation unchanged. Because of low gain (e.g., ~3× per 7mm in a conductive-cooled Yb:YAG [29]), the population inversion amplifier containing energy-level laser media requires a long-length propagation and/or a multiple-pass configuration for high energy amplification, which is not suitable here. OPA based on the three-wave coupling in nonlinear crystals can realize high gain within a thin crystal (e.g., 20× per 0.5mm in a 40 fs typeI-BBO OPA [30] and 10⁶× per 15mm in a 3 ps typeI-BBO OPA [31]), which might be an ideal amplifier for ST wave packets. In this article, we report, under the narrowband approximation, the ST spectrum (angle-dependent phase-matching spectrum) of the type-I collinear OPA has the same analytical expression with that of ST wave packets, which accordingly is a natural amplifier for ST wave packets. The amplified signal keeps the ST correlation of the incident ST wave packet, and the simultaneously generated idler also has high ST correlation which is another ST wave packet, forming a collinear-propagating double ST wave packet. In the non-degenerate OPA, two ST wave packets within the double ST wave packet at short- and long-wavelengths (relative to the degenerate-wavelength) have superluminal and subluminal velocities, respectively; and in the degenerate OPA, it becomes a luminal quasi-monochromatic (the degenerate-wavelength) plane wave pulse. Our numerical simulation verified the theoretical prediction and could also be applied in experiments.

Figure 1(a) shows in the Fourier space (momentum space) k_x-k_z-k of the physical space (position space) x-z-t, where x and z are the transverse and longitudinal space axes and t is the time axis, the ST spectrum of a ST wave packet lies in the intersecting curve between the light-cone and a spectral plane parallel to the k_x-axis [8, 9, 12–25], and satisfies

$$\cos \Delta \theta =\frac{{{\omega _0}+\Omega \cot \alpha }}{{{\omega _0}+\Omega }}$$

where, Δθ is the spectral-dependent propagating angle with respect to that of ω₀, ω₀ is the vertex frequency of the ST spectrum defined as the intersecting point between the ST spectrum and the plane k_x = 0 at k_z > 0 and has the coordinates (k_x, k_z, ω/c) = (0, k₀, ω₀/c), Ω = ω – ω₀ is the frequency difference, and α is the tilt angle of the spectral plane with respect to the k_z-axis, which determines the velocity of a ST wave packet v_g/c = tanα [13, 23]. Figure 1(a) illustrates a superluminal and a subluminal ST wave packets with α₁ > 45º and α₂ < 45º at two different vertex frequencies, respectively.

Figure 1(b) shows in the type-I (o + o → e) collinear OPA, three waves propagate along the z-axis, and the energy is transferred from the strong pump (e-light) to the amplified signal (o-light) and the generated idler (o-light) in a thin nonlinear crystal (BBO here). We define x-y-z and x’-y’-z’ are the lab and crystal coordinates, respectively; θ_s, θ_i and θ_p are propagating angles of the signal, idler and pump in the x’-y’-z’ coordinates; because the pump always propagates along the z-axis, Δθ_s = θ_s - θ_p and Δθ_i = θ_i - θ_p are propagating angles of the signal and idler in the x-y-z coordinates. The relative rotation angle between the x-y-z and x’-y’-z’ coordinates about the y/y’-axis is θ_p and changes the refractive index of the pump (e-light). Traditional collinear and non-collinear phase-matchings (momentum conservation) and energy conservation are illustrated in Fig. 1(c). If a ST wave packet can satisfy the amplification condition, Fig. 1(c) shows, apart from energy conversion, whose phase-matching should contain a collinear phase-matching for the vertex frequency ω₀ and a series of spatially symmetric non-collinear phase-matchings for other frequencies ω = ω₀ + Ω. Using a 515 nm pump, phase-matching curves for different pump angles θ_p from 23.02º to 23.42º every 0.1º are calculated and shown in Fig. 1(d). Here, we use solid and dash curves denote the signal and idler, respectively. The pump is always along k_x = 0, for the collinear phase-matching, both the signal and idler are along k_x = 0; and for non-collinear phase-matchings, the signal and idler are at either side of the pump. Comparing the angle-dependent phase-matching spectrum (here also named as ST spectrum) of the signal or idler [see Fig. 1(d)] with the ST spectrum of ST wave packets [see Fig. 1(a)], they have quite similar profiles in the λ-k_x (ω/c-k_x) plane, i.e., continuous symmetric conic curves about k_x = 0. If two curves overlap with each other, the vertex frequency ω₀ and other frequencies ω = ω₀ + Ω of a ST wave packet would be simultaneously amplified by collinear and non-collinear OPAs, respectively, generating an idler that is also a ST wave packet. The amplified signal and the generated idler would have different frequencies for the non-degenerate OPA and the same vertex frequency for the degenerate OPA, respectively.

To verify this judgment, we derived the analytical expression of the angle-dependent phase-matching spectrum (ST spectrum) of OPA under the narrowband approximation, which satisfies

$$\cos \Delta {\theta _s} \approx \frac{{{\omega _{s0}}+{\Omega _s}{{\left( {{\omega _{s0}}n_{s}^{2}+\frac{{{{\left( {{\omega _p}{n_p} - {\omega _{s0}}{n_s}} \right)}^2}}}{{{\omega _p} - {\omega _{s0}}}}} \right)} \mathord{\left/ {\vphantom {{\left( {{\omega _{s0}}n_{s}^{2}+\frac{{{{\left( {{\omega _p}{n_p} - {\omega _{s0}}{n_s}} \right)}^2}}}{{{\omega _p} - {\omega _{s0}}}}} \right)} {{\omega _p}{n_p}{n_s}}}} \right. \kern-0pt} {{\omega _p}{n_p}{n_s}}}}}{{{\omega _{s0}}+{\Omega _s}}}$$

where, Δθ_s is the propagating angle of the frequency ω_s = ω_s0 + Ω_s with respective to the vertex frequency ω_s0 within the signal ST wave packet in the x-y-z coordinates, n_s and n_p are refractive indices of the vertex frequency ω_s0 and the pump frequency ω_p, and constant n_s is used for all frequencies of the signal under the narrowband approximation. By comparing Eqs. (2) with (1), the ST spectrum (angle-dependent phase-matching spectrum) of the signal in OPA has the same expression with that of ST wave packets. This indicates that a ST wave packet can be perfectly amplified by a matched OPA. The tilt angle α_s of the spectral plane in the Fourier space is given by

$$\cot {\alpha _s} \approx \frac{{{\omega _{s0}}n_{s}^{2}+\frac{{{{\left( {{\omega _p}{n_p} - {\omega _{s0}}{n_s}} \right)}^2}}}{{{\omega _p} - {\omega _{s0}}}}}}{{{\omega _p}{n_p}{n_s}}}$$

The refractive indices n_s and n_p in the crystal can be calculated by the Sellmeier equation for ω_s0 and ω_p, respectively, and the pump angle θ_p is contained in n_p. Because three-wave coupling equations show the signal and idler are symmetric in mathematics [see Eq. (5) in Methods], the generated idler is also a ST wave packet, whose ST spectrum Δθ_i(Ω_i) and spectral plane tilt angle α_i are also described by Eqs. (2) and (3) by simply replacing the subscript s by i.

Solving the collinear phase-matching [Eq. (5) in Methods] in a 515 nm pumped type-I BBO crystal, at different pump angles θ_p, two phase-matched vertex wavelengths λ₀ of the signal and idler in a double ST wave packet can be calculated, and by substituting into Eq. (3), their velocities are obtained (v_g/c = tanα) and shown in Fig. 2. A signal/idler and its idler/signal lie in red and black curves, respectively. When the pump angle is θ_p < 23.32º or θ_p = 23.32º, OPA is nondegenerate or degenerate, respectively, and the double ST wave packet has “different wavelengths and velocities” or “same wavelength and velocity”, respectively. In the nondegenerate OPA, velocities of a double ST wave packet are superluminal and subluminal when the vertex wavelengths are shorter and longer than the degenerate-wavelength (1030 nm here), respectively. In this case, the two ST wave packets will separate in time during propagation, which can also be separated in spectrum. In the degenerate OPA, the signal and idler have a same vertex wavelength of the degenerate-wavelength (1030 nm here), a same 45º tilt angle of spectral planes in the Fourier space [see Eq. (3)], and a same velocity of the light speed (as introduced in Refs. [13, 25], ST wave packets become luminal plane wave pulses). However, Fig. 1(d) shows only the vertex wavelength satisfies OPA, and the amplified signal and the generated idler become quasi-monochromatic (the degenerate-wavelength) luminal plane wave pulses.

In addition to the strict phase-matching (Δk = 0), small phase-mismatchings (Δk → 0) can also support OPAs. Red zones in Figs. 3(a) and 3(b) show gain spectra in a 2 mm thin type-I BBO crystal, when the pump wavelength, angle, and intensity are 515 nm, θ_p = 23º, and 10 GW/cm², respectively. Although two gain spectra in both short- and long-wave ranges have some ST correlations [see Figs. 3(a) and 3(b)], Figs. 3(c) and 3(d) show the supported pulsed beams have Gaussian ST distribtuions, which accordingly are not diffraction-free. H. E. Kondakci et al. have defined a parameter of the correlation uncertainty δλ between k_x and λ, which is inversely proportional to the propagation-invariant length [8]. Large correlation uncertainties δλ = 43 and 99 nm of short- and long-wave gain spectra distort ST correlations, and that is why no diffraction-free beam has been previously reported in OPA, optical parametric generation (OPG), or optical parametric oscillation (OPO). A ST wave packet is optimized [using Eq. (3)] for seeding the above OPA, whose ST spectrum is shown by the yellow zone in Fig. 3(a), and parameters include a reduced correlation uncertainty δλ = 2 nm, a vertex wavelength λ₀ = 855 nm, a bandwidth Δλ = 40 nm, and a tilt angle of the spectral plane in the Fourier space α = 45.0572º. The ST spectrum of the generated idler is illustrated by the yellow zone in Fig. 3(b), and parameters are a correlation uncertainty δλ = 4.7 nm, a vertex wavelength λ₀ = 1295 nm, a bandwidth Δλ = 105 nm, and a tilt angle of the spectral plane in the Fourier space α = 44.9111º. The 10 GW/cm² strong pump enables 400× gain for the signal and generates an idler of the same energy level. Figures 3(e) and 3(f) show the pulsed beams supported by the ST spectra of the signal and idler, and both have a typical distribution of ST wave packets with a butterfly-like shape and a central peak [8, 9]. Although the “central peaks” are enlarged in space and time [compare Figs. 3(e, f) and 3(c, d)], which do not spread out during long-distance propagation much exceeding the Rayleigh length. We can also find some small mis-matchings between ST spectra of signal and idler ST wave packets and strict phase-matching curves (dash lines) in Figs. 3(a) and 3(b), especially when wavelengths are away from the vertexes. It is because the narrowband approximation for Eq. (2) cannot be strictly satisfied. However, ST spectra of signal and idler ST wave packets still lie in gain spectra of OPA, which doesn’t affect amplification. In experiments, a near-perfect phase-matching can be obtained by reducing both the bandwidth Δλ and the correlation uncertainty δλ of the signal ST wave packet.

The free propagation of the amplified signal and the generated idler in vacuum are shown in Fig. 4. OPA happens at the position z = 0, where the signal and idler overlap in space and time. After z = 0, the separation between two ST wave packets increases with propagation due to different velocities of the signal and idler v_gs = 1.0020c and v_gi = 0.9969c. Figure 4 also shows the propagation-invariant length of the idler is shorter than that of the signal. The first reason is the Rayleigh length for a fixed beam waist w₀ decreases with increasing the wavelength (e.g., Gaussian beam Z_R = πw₀²/λ). Here, both central-peak waists of signal and idler ST wave packets are around w₀ = 23.5 µm, however due to a longer wavelength (1295 and 855 nm vertex wavelengths for the idler and signal), the Rayleigh length of the idler Z_Ri = 1.3 mm is shorter than that of the signal Z_Rs = 2.0 mm. The second reason is the correlation uncertainty of the idler δλ = 4.7 is larger than that of the signal δλ = 2 and accordingly shortens the propagation-invariant length. Even so, Fig. 4 shows both the propagation-invariant lengths have far exceeded their Rayleigh lengths of Z_Rs = 2.0 mm and Z_Ri = 1.3 mm.

In summary, we have proposed a principle of amplifying a ST wave packet to high-energy by the type-I collinear OPA, where an idler ST wave packet would be generated simultaneously. Our numerical simulation verified the feasibility of this principle and demonstrated an as yet unreported strong double ST wave packet. In the nondegenerate OPA, two collinear-propagating ST wave packets at short and long wavelengths (compared with the degenerate-wavelength) are superluminal and subluminal, respectively, which accordingly separate in time during propagation. In the degenerate OPA, the double ST wave packet becomes a quasi-monochromatic (the degenerate-wavelength) luminal plane wave pulse. Amplifying ST wave packets in energy will introduce this new type light source to strong-field optics for studying ST-light–matter interactions. Moreover, simultaneously generated idlers can turn ST wave packets to other wavelengths, such as infrared, visible and ultraviolet wavelengths, and switch velocities between superluminal and subluminal. This discovery will expand the range of applications of ST wave packets.

ST spectrum of ST wave packet. In the Fourier space k_x-k_z-k, the ST spectrum of a ST wave packet is governed by the simultaneous equations

$$k_{x}^{2}+k_{z}^{2}={k^2}$$

$$k - {k_0}=\left( {{k_z} - {k_0}} \right)\tan \alpha$$

where, k = ω/c (in the vacuum) is the wavenumber, k_x = k⋅sinΔθ and k_z = k⋅cosΔθ are the transverse and longitudinal components, Δθ is the spectral-dependent propagating angle with respect to that of the vertex frequency, and k₀ is the vertex wavenumber of the ST spectrum defined as the intersecting point between the ST spectrum and the plane k_x = 0 at k_z > 0 and has the coordinates (k_x, k_z, k) = (0, k₀, k₀). α is the tilt angle of the spectral plane with respect to the k_z-axis. By substituting Eq. (4b) into Eq. (4a) and using Ω = ω – ω₀, |Δθ| < π/2, Ω/ω₀ < 1 and α ~ π/4, we get the ST spectrum of a ST wave packet that is given by Eq. (1) [see Supplement 1 in Supplementary information].

ST spectrum of OPA. The three-wave coupling among the pump, signal and idler satisfies energy conservation and momentum conservation (phase-matching)

$${\omega _s}+{\omega _i}={\omega _p}$$

$${\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {k} _s}+{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {k} _i}={\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {k} _p}$$

where, $\stackrel{⃑}{k}$ is the wavevector in the nonlinear crystal, and the subscript s, i, and p denotes the signal, idler and pump, respectively. The vertex frequency ω_s0 of the signal ST wave packet satisfies the collinear phase-matching (Δθ_s0 = 0), and Eq. (5b) becomes ω_s0n_s + ω_i0n_i = ω_pn_p, where ω_i0 is the idler of ω_s0, and n_s, n_i and n_p are refractive indices of ω_s0, ω_i0 and ω_p. By substitution of Eq. (5a), n_i is given by

$${{\text{n}}_i}=\frac{{{\omega _p}{n_p} - {\omega _{s0}}{n_s}}}{{{\omega _p} - {\omega _{s0}}}}$$

Because the bandwidth of the signal ST wave packet, accordingly that of the generated idler, usually is narrow Ω/ω₀ << 1, we can approximately use constant refractive indices n_s and n_i for all frequencies of the signal and idler, respectively. Other frequencies ω_s = ω_s0 + Ω_s of the signal ST wave packet satisfy the non-collinear phase-matchings (Δθ_s ≠ 0), and Eq. (5b) becomes

$$\omega _{i}^{2}n_{i}^{2}=\omega _{p}^{2}n_{p}^{2}+\omega _{s}^{2}n_{s}^{2} - 2{\omega _p}{n_p}{\omega _s}{n_s}\cos \Delta {\theta _s}$$

where, Δθ_s is the non-collinear angle between the signal and pump, which is also the propagating angle of the frequency ω_s = ω_s0 + Ω_s with respective to the vertex frequency ω_s0 within the signal ST wave packet in the x-y-z coordinates. By substituting Eqs. (5a) and (6) into Eq. (7), replacing ω_s by ω_s0 + Ω_s, and using the narrowband approximation Ω_s² ≈ 0, we get the angle-dependent phase-matching spectrum (ST spectrum) of OPA that is given by Eq. (2) [see Supplement 2 in Supplementary information].

Gain spectrum of OPA and amplified intensity. Under the undepleted pump approximation, the gain G of OPA as well as the output intensities of the signal I_s(L) and idler I_i(L) are given by [32]

$$G=\left\{ {\begin{array}{*{20}{c}} {\frac{{{\Gamma ^2}}}{{{\Gamma ^2} - {{\left( {\frac{{\Delta k}}{2}} \right)}^2}}}{{\sinh }^2}\left[ {L\sqrt {{\Gamma ^2} - {{\left( {\frac{{\Delta k}}{2}} \right)}^2}} } \right],\begin{array}{*{20}{c}} {}&{} \end{array}\begin{array}{*{20}{c}} {for}&{{\Gamma ^2} \geqslant {{\left( {\frac{{\Delta k}}{2}} \right)}^2}} \end{array}} \\ {\frac{{{\Gamma ^2}}}{{{{\left( {\frac{{\Delta k}}{2}} \right)}^2} - {\Gamma ^2}}}{{\sin }^2}\left[ {L\sqrt {{{\left( {\frac{{\Delta k}}{2}} \right)}^2} - {\Gamma ^2}} } \right],\begin{array}{*{20}{c}} {}&{} \end{array}\begin{array}{*{20}{c}} {for}&{{\Gamma ^2}<{{\left( {\frac{{\Delta k}}{2}} \right)}^2}} \end{array}} \end{array}} \right.$$

$${\Gamma ^2}=\frac{{2{\omega _s}{\omega _i}d_{{eff}}^{2}{I_p}}}{{{n_p}{n_s}{n_i}{\varepsilon _o}{c^3}}}$$

$${I_s}\left( L \right)={I_s}\left( 0 \right) \cdot \left( {1+G} \right)$$

$${I_i}\left( L \right)={I_s}\left( 0 \right)\frac{{{\omega _i}}}{{{\omega _s}}}G$$

where, L is the crystal length, d_eff is the effective nonlinear coefficient, and I_p is the undepleted strong pump intensity.

Propagation of double ST wave packet. The angular spectrum method is used to simulate free propagation of the double ST wave packet in vacuum, which from the position z = 0 to the position z is governed by

$$E\left( {x,z,t} \right)={\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 {2\pi }}}\right.\kern-0pt}\!\lower0.7ex\hbox{${2\pi }$}}\int {E\left( {x,z,\omega } \right)\exp \left( {i\omega t} \right)d\omega }$$

$$E\left( {x,z,\omega } \right)={\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 {2\pi }}}\right.\kern-0pt}\!\lower0.7ex\hbox{${2\pi }$}}\int {A\left( {{k_x},z,\omega } \right)\exp \left( {i{k_x}x} \right)d{k_x}}$$

$$A\left( {{k_x},z,\omega } \right)=A\left( {{k_x},z=0,\omega } \right)\exp \left( {ikz\sqrt {1 - {{k_{x}^{2}} \mathord{\left/ {\vphantom {{k_{x}^{2}} {{k^2}}}} \right. \kern-0pt} {{k^2}}}} } \right)$$

$$A\left( {{k_x},z=0,\omega } \right)=\int {E\left( {x,z=0,\omega } \right)\exp \left( { - i{k_x}x} \right)dx}$$

$$E\left( {x,z=0,\omega } \right)=\int {E\left( {x,z=0,t} \right)\exp \left( { - i\omega t} \right)dt}$$

Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Acknowledgements

This work was supported by Zhangjiang Laboratory.

Author contributions

Z.L. developed the concept, derived the equations, carried out the simulation, and prepared the manuscript. Y.L. and R.L. optimized the concept, revised the manuscript, and discussed the application. All authors discussed the results and commented on the manuscript.

Competing interests

The authors declare no competing interests.

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